My Sum on Cevian Theorem

"In Δ ABC, B-P-C, BP = 4, PC = 8, A-Q-C, AQ = QC = 6. BQ intersects AP at point X. If CX is drawn and extended to intersect AB at point R , find length of PR.

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Triple Heronian Triangle with all the three integer sides and all the three corresponding integer heights (altitudes)

Is it possible to have a triangle with all three integer sides with their corresponding heights ( altitudes ) also integers. the answer is very much yes. I am dedicating this post to the memory of my parents, who selflessly loved and brought me up. Also due to the grace of Shree Akkalkot Swami Maharaj, Mahrashtra, India and Shree Gajanan MAharaj, Shegaon, MH, India I am able to make this post. Please find attached photos of MS Excel file that are self-explanatory. Actually I had lost the measures of the triangle that I made three years ago; but after praying to my Saints, now I can explain a method to form such many triangles. I am thankful to my above mentioned Sadguru Saints for this post. Please comment, suggest and correct me if I am wrong. Hemant Dikshit

Similarity Extra or HOTS sums

How to find quickly square root of a perfect square polynomial of degree 6?

How to find quickly square root of a perfect square polynomial of degree 6? This is a little method which will enable you to get the answer quickly. Let the polynomail be Px^6+Qx^5+Rx^4+Sx^3+Tx^2+Ux+V. Let its root be ax^3+bx^2+cx+d. a= square root of P. b= Q/2a c= (R-b^2)/2a d= U/2c

Coordinate geometry trick: If co-ordinates of two points on a line are given and you are asked to find the equation of the line. How to find it quickly.

 Coordinate geometry trick:

If co-ordinates of two points on a line are given and you are asked to find the equation of the line. How to find it quickly.

0) If Y2/X2 = Y1/X1 = k

the required equation is y = kx.

or kx-y=0.

Example 1:

A (2,4)

B (1,2)

Here it is quite easy to understand that k = 4/2 = 2/1 = 2.

so y = 2x

or 2x-y=0 is the required equation of the line.

*Or*

1) Write first the coordinates of that point whose y- co-ordinate is larger than y-co-ordinate of the other point.

2) Below it, write the co-ordinates of the other point.

3) Find the difference between the Y- co-ordinates. You have got now the coefficient of the x in the required equation of the line.

4) find the negative of the difference between the X coordinates in the same order. This is the coefficient of y in the required equation of the line.

5) Now X2 , Y2

              X1 , Y1.

Find + ( X2 Y1 - X1 Y2 ).

It is the constant part of the equation.

5a) write the answers obtained steps 3), 4) and 5) with x, y and appropriate signs for the constant, and equate it to zero.

6) This is the required equation of the line in a x + b y + c = 0 form. *Refer Example 2 given below.*

7) If in a x + b y + c = 0, a, b & c not coprimes, then divide the equation by the highest common factor (HCF) of a,b,c.

This is your required equation. *Refer Example 3 given below.*

Example 2:

A(3,-2)

B(4,-4)

-2>-4 

A(3,-2)

B(4,-4)

[-2-(-4)] x - (3-4) y + (3x-4 - 4x-2) =0

2x+y+(-12+8)=0

2x+y=4.

is the required equation.

Example 3:

B (1,-1)

A (7,7)

7 > -1

A (7,7)

B (1,-1)

[(7 - (-1) ] x -(7 - 1)y + (7 x - 1 - 7 x 1) = 0

8X - 6y -14 =0

8, 6 and 14 are not coprimes and their HCF is 2 therefore dividing the equation by 2 we get,

4x -3y - 7 = 0

or 4x -3y = 7 is the required equation of the line passing through the given points.

Hope you liked the post. It is more useful for 11 Std Science student who are preparing for CET or JEE.

How to form sums based on Apollonius theorem in which 3 sides of a triangle and a median all are natural numbers?

1) We know that, an odd number's square will be odd. Adding two odd numbers square will give you even number. So adding squares of two odd numbers will give you an even number.

2) Take 19 and other prime number.

3) Add their squares. Divide it by 2.

4) Find in how many ways the last digit can be formed. 

5) Try to express the number obtained in step 3 as square of two numbers. You may write natural numbers and their squares separately.

6) Remember 2,3,7 and 8 do not come at the end of (unit place of) a perfect square.

7) Discard those possibilities.

8) 1,3,5,7 and 9 can come at / as last digit in following ways.

1: 0 & 1 or 5 & 6

3: 4 & 9

5: 1 & 4; 5 & 0 or 6 & 9

7: 1 & 6

9: 4 & 5 or 0 & 9.

For above mentioned step 5) consider natural numbers up to square root of the largest perfect square less than or equal the sum obtained step 3.

9) We know that sum of two sides of a triangle is greater than third side.

Check whether all the three triangles so formed satisfy this condition.

10) You can create another sum. If three sides of larger triangle are obtained as a-b-2d and median on side 2d as c; the other sum (triangle) is a-b-2c and its median will be d.

- Thursday, the midnight of 22 Nov 18 2:12 a.m. 23 Nov 2018.

3 Prostrations to Shree Saibaba, Shirdi, Dist. Ahmednagar, Maharashtra, India!!!

For example:

In triangle ABC, sides are as given below. Find length of median AM on side BC.

1) AB=13, AC=29, BC=24.

2) AB=13, AC=29, BC=38.